The constant function f(x) = 0 will give zero no matter what function
g(x) it is integrated with. Does this mean that the constant function
zero is orthogonal to all functions? Also, what could be the geometrical
interpretation of orthogonal functions?

I know the area of a sphere is 4phi(r^2), but I'm wondering how to
derive that formula. I know it should be done in cylindrical
coordinates, and I'm thinking that the arc of a circle is defined as
rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi).
Could you please help explain this?

I am looking for an analytical method, or a proof that an analytical
method does not exist, for the following: 1. Partial Surface Area of an
Ellipsoid, 2. Partial Volume of an Ellipsoid, 3. The Intersection Volume
of Two or More Ellipsoids.